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A Voronovskaya Theorem for Variation-Diminishing Spline Approximation

Published online by Cambridge University Press:  20 November 2018

M. J. Marsden
M. J. Marsden*
Affiliation:
University of Pittsburgh, Pittsburgh, Pennsylvania
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In [7] Schoenberg introduced the following variation-diminishing spline approximation methods.

Let m > 1 be an integer and let Δ = {xi} be a biinfinite sequence of real numbers with xi ≧ xi + l < xi+m. To a function f associate the spline function Vf of order m with knots Δ defined by

(1.1)

where

and the Nj(x) are B-splines with support xj < x < xj+m normalized so that ΣjNj(x) = 1. See, e.g., [2] for a precise definition of the Nj(x) and a discussion of the properties of Vf.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 5 , 01 October 1986 , pp. 1081 - 1093
Copyright
Copyright © Canadian Mathematical Society 1986

References

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6. Marsden, M. J. and Riemenschneider, S. D., Asymptotic formulae for variation-diminishing splines, in second Edmonton Conference on Approximation Theory, CMS/AMS Conference Proceedings 3 (American Mathematical Society, Providence, 1983), 255–261.Google Scholar
7. Schoenberg, I. J., On spline functions, In Inequalities (Academic Press, New York, 1967), 255–291.Google Scholar